151505
20190509132331.0
10.5075/epfl-thesis-4871
doi
urn:nbn:ch:bel-epfl-thesis4871-1
urn
6153916
nebis
THESIS
eng
4871
Signal Processing in Space and Time
A Multidimensional Fourier Approach
Lausanne
2010
EPFL
2010
174
Theses
Sound waves propagate through space and time by transference of energy between the particles in the medium, which vibrate according to the oscillation patterns of the waves. These vibrations can be captured by a microphone and translated into a digital signal, representing the amplitude of the sound pressure as a function of time. The signal obtained by the microphone characterizes the time-domain behavior of the acoustic wave field, but has no information related to the spatial domain. The spatial information can be obtained by measuring the vibrations with an array of microphones distributed at multiple locations in space. This allows the amplitude of the sound pressure to be represented not only as a function of time but also as a function of space. The use of microphone arrays creates a new class of signals that is somewhat unfamiliar to Fourier analysis. Current paradigms try to circumvent the problem by treating the microphone signals as multiple "cooperating" signals, and applying the Fourier analysis to each signal individually. Conceptually, however, this is not faithful to the mathematics of the wave equation, which expresses the acoustic wave field as a single function of space and time, and not as multiple functions of time. The goal of this thesis is to provide a formulation of Fourier theory that treats the wave field as a single function of space and time, and allows it to be processed as a multidimensional signal using the theory of digital signal processing (DSP). We base this on a physical principle known as the Huygens principle, which essentially says that the wave field can be sampled at the surface of a given region in space and subsequently reconstructed in the same region, using only the samples obtained at the surface. To translate this into DSP language, we show that the Huygens principle can be expressed as a linear system that is both space- and time-invariant, and can be formulated as a convolution operation. If the input signal is transformed into the spatio-temporal Fourier domain, the system can also be analyzed according to its frequency response. In the first half of the thesis, we derive theoretical results that express the 4-D Fourier transform of the wave field as a function of the parameters of the scene, such as the number of sources and their locations, the source signals, and the geometry of the microphone array. We also show that the wave field can be effectively analyzed on a small scale using what we call the space/time-frequency representation space, consisting of a Gabor representation across the spatio-temporal manifold defined by the microphone array. These results are obtained by treating the signals as continuous functions of space and time. The second half of the thesis is dedicated to processing the wave field in discrete space and time, using Nyquist sampling theory and multidimensional filter banks theory. In particular, we show examples of orthogonal filter banks that effectively represent the wave field in terms of its elementary components while satisfying the requirements of critical sampling and perfect reconstruction of the input. We discuss the architecture of such filter banks, and demonstrate their applicability in the context of real applications, such as spatial filtering and wave field coding.
acoustic wave fields, array signal processing
multidimensional Fourier analysis
space/time-frequency analysis
multidimensional filter banks
directional filter banks
spatial filtering
wave field coding
champs d'ondes acoustiques
traitement des signaux réseaux
analyse de Fourier multidimensionnelle
analyse espace/temps-fréquence
bancs de filtres multidimensionnels
bancs de filtres directionnels
filtrage spatial
codification du champ d'ondes
Pinto, Francisco Pereira Correia
Vetterli, Martin
dir.
107537
240184
Texte intégral / Full text
15546232
Texte intégral / Full text
http://infoscience.epfl.ch/record/151505/files/EPFL_TH4871.pdf
LCAV
252056
U10434
oai:infoscience.tind.io:151505
IC
DOI
thesis-bn2018
thesis
GLOBAL_SET
DOI2
IC
ISC
EDIC2005-2015
LCAV1
2010
4871/THESES
EPFL
PUBLISHED
THESIS